Introduction |
Notation |
Local Inversion / 1: |
A Preliminary Statement / 1.1: |
Partial Derivatives. Strictly Differentiable Functions / 1.3: |
The Local Inversion Theorem: General Statement / 1.4: |
Functions of Class Cr / 1.5: |
The Local Inversion Theorem for Cr maps / 1.6: |
Generalizations of the Local Inversion Theorem / 1.8: |
Submanifolds / 2: |
Definitions of Submanifolds / 2.1: |
First Examples / 2.3: |
Tangent Spaces of a Submanifold / 2.4: |
Transversality: Intersections / 2.5: |
Transversality: Inverse Images / 2.6: |
The Implicit Function Theorem / 2.7: |
Diffeomorphisms of Submanifolds / 2.8: |
Parametrizations, Immersions and Embeddings / 2.9: |
Proper Maps: Proper Embeddings / 2.10: |
From Submanifolds to Manifolds / 2.11: |
Some History / 2.12: |
Transversality Theorems / 3: |
Countability Properties in Topology / 3.1: |
Negligible Subsets / 3.3: |
The Complement of the Image of a Submanifold / 3.4: |
Sard''s Theorem / 3.5: |
Critical Points, Submersions and the Geometrical Form of Sard''s Theorem / 3.6: |
The Transversality Theorem: Weak Form / 3.7: |
Jet Spaces / 3.8: |
The Thom Transversality Theorem / 3.9: |
Classification of Differentiable Functions / 3.10: |
Taylor Formulae Without Remainder / 4.1: |
The Problem of Classification of Maps / 4.3: |
Critical Points: the Hessian Form / 4.4: |
The Morse Lemma / 4.5: |
Fiburcations of Critical Points / 4.6: |
Apparent Contour of a Surface in R3 / 4.7: |
Maps from R2 into R2 / 4.8: |
Envelopes of Plane Curves / 4.9: |
Caustics / 4.10: |
Genericity and Stability / 4.11: |
Catastrophe Theory / 5: |
The Language of Germs / 5.1: |
r-sufficient Jets; r-determined Germs / 5.3: |
The Jacobian Ideal / 5.4: |
The Theorem on Sufficiency of Jets / 5.5: |
Deformations of a Singularity / 5.6: |
The Principles of Catastrophe Theory / 5.7: |
Catastrophes of Cusp Type / 5.8: |
A Cusp Example / 5.9: |
Liquid-Vapour Equilibrium / 5.10: |
The Elementary Catastrophes / 5.11: |
Catastrophes and Controversies / 5.12: |
Vector Fields / 6: |
Exemples of Vector Fields (Rn Case) / 6.1: |
First Integrals / 6.3: |
Vector Fields on Submanifolds / 6.4: |
The Uniqueness Theorem and Maximal Integral Curves / 6.5: |
One-parameter Groups of Diffeomorphisms / 6.6: |
The Existence Theorem (Local Case) / 6.8: |
The Existence Theorem (Global Case) / 6.9: |
The Integral Flow of a Vector Field / 6.10: |
The Main Features of a Phase Portrait / 6.11: |
Discrete Flows and Continuous Flows / 6.12: |
Linear Vector Fields / 7: |
The Spectrum of an Endomorphism / 7.1: |
Space Decomposition Corresponding to Partition of the Spectrum / 7.3: |
Norm and Eigenvalues / 7.4: |
Contracting, Expanding and Hyperbolic Endommorphisms / 7.5: |
The Exponential of an Endomorphism / 7.6: |
One-parameter Groups of Linear Transformations / 7.7: |
The Image of the Exponential / 7.8: |
Contracting, Expanding and Hyperbolic Exponential Flows / 7.9: |
Topological Classification of Linear Vector Fields / 7.10: |
Topological Classification of Automorphisms / 7.11: |
Classification of Linear Flows in Dimension 2 / 7.12: |
Singular Pints of Vector Fields / 8: |
The Classification Problem / 8.1: |
Linearization of a Vector Field in the Neighbourhodd of a Singular Point / 8.3: |
Difficulties with Linearization / 8.4: |
Singularities with Attracting Linearization / 8.5: |
Liapunov Theory / 8.6: |
The Theorems of Grobman and Hartman / 8.7: |
Stable and Unstable Manifolds of a Hyperbolic Singularity / 8.8: |
Differentiable Linearization: Statement of the Problem / 8.9: |
Differentiable Linearization: Resonances / 8.10: |
Differentiable Linearization: The Theorems of Sternberg and Hartman / 8.11: |
Linearization in Dimenension 2 / 8.12: |
Some Historical Landmarks / 8.13: |
Closed Orbits - Structural Stability / 9: |
The Poincarè Map / 9.1: |
Characteristic Multipliers of a Closed Orbit / 9.3: |
Attracting Closed Orbits / 9.4: |
Classification of Closed Orbits and Classification of Diffeomorphisms / 9.5: |
Hyperbolic Closed Orbits / 9.6: |
Local Structural Stability / 9.7: |
The Kupka-Smale Theorem / 9.8: |
Morse-Smale Fields / 9.9: |
Structural Stability Through the Ages / 9.10: |
Bifurcations of Phase Portrait / 10: |
What Do We Mean by a Bifurcation? / 10.1: |
The Centre Manifold Theorem / 10.3: |
The Saddle-Node Bifurcation / 10.4: |
The Hopf Bifurcation / 10.5: |
Local Bifurcations Carried by a Closed Orbit / 10.6: |
Saddle / 10.7: |